No. With only the Y component of force you do not have enough information to solve the problem. You must use Newton's second law and then break the problem down into componets in order to find the tension vector.

The technique I posted is the proper way to solve the problem so you will conceptually understand what is going on with all of the forces.

The components depend on the referenced axis and the direction of the angle. Is θ1 = 27.42° from the positive x axis or negative x axis (look at your drawing). What about θ2 = 67.35° referenced position (they should be opposite given your problem but maybe not)?

If θ1 is a negative angle from the negative x axis and θ2 from the positive you have:

x components: -T1cos(θ1) + T2cos(θ2) - 0 = 0

y components: T1sin(θ1) + T2sin(θ2) - Mg = 0

Solve those two equations simultaneously and then substitute back into the equilibrium equation.

Now you have the two equations (assuming the referenced angles were the same as I posted earlier). Solve this one for a numeric value of T1 since you have all the missing information for it. Then plug that answer into the T2 = T1(cos(θ1)/cos(θ2)) to find the other tension.